Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
For Fall/Spring tutorials, see this page.
Archived Summer Tutorials: 2018 2017 2016 2015 2014 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001

Welcome Message

To enroll in a summer tutorial, please email Cindy Jimenez ( by the evening of Monday, May 22 2019, giving her an ordered preference list for the tutorials you wish to enroll in. Note that you can enroll in more than one tutorial if space is available.
An overview of the tutorials is given on Wednesday, May 1, 4 PM in SC 507.

The summer tutorial program offers some interesting mathematics to those of you who will be in the Boston area during July and August. The tutorials will run for six weeks, meeting twice or three times per week in the evenings (so as not to interfere with day time jobs). The first tutorial (Topological Data Analysis) will start early in July or late in June, and run to mid August. The precise starting dates and meeting times will be arranged for the convenience of the participants once the tutorial roster is set. The tutorial on Elliptic Curves will start on July 15 and run till the last week in August. The tutorial on knot invariants starts in early June and runs until mid or late July.
The format of each tutorial will be much like that of the term-time tutorials, with the tutorial leader lecturing in the first few meetings and students presenting later on. Unlike the term-time tutorials, the summer tutorials have no official Harvard status: you will not receive either Harvard or concentration credit for them. Moreover, enrollment in the tutorial does not qualify you for any Harvard-related perks (such as a place to live). However, the Math Department will pay each Harvard College student participant a stipend, and you can hand in your final paper from the tutorial for your junior paper requirement for the Math Concentration.
The topics and leaders of the tutorials this summer are:

Topological Data Analysis

Taught by Jun-Hou Fung Topological data analysis (TDA) is one of the hottest new developments in visualizing and interpreting data. TDA brings together ideas from category theory, algebraic topology, representation theory, metric geometry, probability theory, and statistics to find and analyze patterns in complex high-dimensional data sets. One key tool in this subject is persistent homology, which is a way to study the shape of data simultaneously at multiple feature scales. In this tutorial, we will introduce the mathematics underpinning this theory from the very basics to the frontiers of research. Topics include simplicial complexes associated to data, persistence modules, structure theory for barcodes, and stability results for persistent homology. If time allows, we may also discuss further topics such as Morse theory, manifold learning, and statistical inference in TDA, depending on the participants' backgrounds and interests. In recent years, TDA flourished in biological applications, especially with genomic data, and we will survey some of these recent applications of TDA. While the tutorial will focus on the mathematical theory, participants are highly encouraged to pursue other related aspects - statistical, computational, applied, or otherwise - of this exciting new field.

Knot Invariants and Categorification

Taught by Morgan Opie and Joshua Wang. We'll begin with an introduction to knot theory via classical knot invariants (genus, unknotting number, slice genus, etc.) with many pictures and examples, including an introduction to the Jones polynomial. The rest of the tutorial will focus on two generalizations of the Jones polynomial, both having a categorical/algebraic flavor. The first will be Khovanov homology, a theory which "categorifies" the Jones polynomial in the same way that singular homology categorifies Euler characteristic. Singular homology is a functor from the category of topological spaces, and it recovers the Euler characteristic of a (nice enough) space. In a similarly way, Khovanov homology will be a functor from a category whose objects are knots, and it recovers the Jones polynomial. The second generalization is to the knot polynomials defined by Reshetikhin and Turaev using ribbon categories, and will be covered at a pace and level of detail appropriate to the participants. We will develop basic theory of categories as needed, focusing on the visual theory of ribbon categories and its relationship to braids and knots. We will make an effort to highlight aspects of category theory that are relevant, but will not belabor formalities. This geometrically motivated introduction will provide exposure useful for those interested in future study of category theory. We do not expect students to have any prior exposure to knot theory or category theory, but having familiarity with basic algebraic topology (in particular, knowing the definition and basic properties of singular homology) will be very helpful.

Elliptic Curves and Beyond

Taught by Yujie Xu: Elliptic curves are central objects in the study of number theory. There have been many famous conjectures inspired by some explicit computations involving elliptic curves, e.g. the Birch-Swinnerton-Dyer conjecture relating the L-functions to the rank of rational points of elliptic curves; Mordell-Weil theorem etc. Elliptic curves also played an important role in the proof of many famous theorems, e.g. the proof of Fermat's last theorem involves proving the modularity of some elliptic curves. In this tutorial, we will first cover the foundations of elliptic curves, and then, if time permits, we will cover some more advanced topics related to elliptic curves such as modular curves and Galois representations, Abelian varieties and their moduli spaces.
Summer Tutorials: 2018 2017 2016 2015 2014 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001

Last update, 4/24/2019
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